Shell interactions for Dirac operators

TL;DR

This paper investigates the self-adjointness of Dirac operators perturbed by measure-valued potentials, especially surface measures, and explores the existence of zero eigenvalue eigenfunctions using trace estimates and Cauchy operator identities.

Contribution

It provides new results on the self-adjointness of Dirac operators with singular surface measure potentials and analyzes zero eigenvalue eigenfunctions.

Findings

Self-adjointness results for Dirac operators with surface measure potentials

Existence of non-trivial zero eigenvalue eigenfunctions

Application of trace estimates and Cauchy operator identities

Abstract

The self-adjointness of $H+V$ is studied, where $H=-i\alpha\cdot\nabla +m\beta$ is the free Dirac operator in $\R^3$ and $V$ is a measure-valued potential. The potentials $V$ under consideration are given by singular measures with respect to the Lebesgue measure, with special attention to surface measures of bounded regular domains. The existence of non-trivial eigenfunctions with zero eigenvalue naturally appears in our approach, which is based on well known estimates for the trace operator defined on classical Sobolev spaces and some algebraic identities of the Cauchy operator associated to $H$.